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3 Answers
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Verb «añadir» is a ditransitive verb, which requires both a direct object (what you add) and an indirect object (what you add it to). It also has a neutral connotation. You add something to something. Period.

Verb «exagerar» is a transitive verb, which requires a direct object (what you enlarge). It also has a non-neutral connotation: You enlarge something excessively (e.g. you exaggerate something).

There are neutral translations to enlarge, such as «agrandar» and «aumentar», which are also transitive, requiring an object (what you enlarge).

1. complicaciones might be changed for a more expressive word, with probable regional variants. In Colombia and the Caribbean that be «vainas».

So, first, «añadir» requires a completely different construction than «exagerar». It requires something to be added and something to add that to. What you add might imply it is “too much“ or might not.

Secondly, the difference between «agrandar» and «exagerar» is the same difference than between “to enlarge” and “to exaggerate”. The first is neutral and the second implies excess.

In these particular examples, «exagerar el problema» would mean that he complains about the problem by claiming that the problem is bigger than it is, or to make the problem bigger than it should be, while «agrandar el problema» is to actually make the problem bigger than it was; or to find arguments to claim that the problem is bigger than it is.

PS.

Thinking in the construction «cada vez cuando» I might find a possibility that it is not completely incorrect, but it would require a comma:

With the comma is still a little stylistic awkward but would be grammatically correct. In this case «cada vez» would mean somthing like «cada vez que habla», while “we working in a project” is the context. I've heard and used that tacit «que habla» but it is not a good style. A better rephrasing of that meaning would be.

What dictionary is that? “Añadir” doesn't mean to exaggerate, unless it's some obscure dialectal usage (in which case I doubt it would appear in a Spanish-English dictionary).

“Exagerar” means to exaggerate (to say something is bigger/better/etc than it actually is), and “añadir” just means to add something to something else, possibly making it bigger.

But your second sentence is strange. Written as is, it means there is a problem everybody knows about, including the reader, and every time we are working on a project he adds it (to the project--it's implied).

If you wanted to say he makes the problem bigger (by adding something to it), then you should say “él añade al problema”.